In Random Dynamical Systems - Arnold Ludwig, Example 3.3.9.
The cocycle generated by a random matrix: $$A=\left(\begin{array}{cc} a & c\\ 0 & b \end{array}\right)$$
is given by:
$$\Phi(n,\omega)=\left(\begin{array}{cc} a_{n-1}\cdots a_{0} & \sum_{k=0}^{n-1}a_{n-1}\cdots a_{k+1}c_kb_{k-1}\cdots b_{0}\\ 0 & b_{n-1}\cdots b_{0} \end{array}\right).$$
In part (iii) the Lyapunov index of the individual elements of $\Phi$ are then determined. Trivially we have $\lambda(\phi_{11}) = \mathbb{E}\log|a|$ and $\lambda(\Phi_{22})=\mathbb{E}\log|b|$, but then the example goes on without any explanation and determines that the index: $$ \lambda(\Phi_{12})\leq\max\{\lambda(\Phi_{11}),\lambda(\Phi_{22})\}.$$
I assume that this somehow follows from:
$$\lambda(f+g)\leq\max\{\lambda(f),\lambda(g)\},$$
but I'm not sure. Can someone enlighten me?
Indeed this hardly needs a justification. One knows that $\prod\limits_{k=1}^na_{n-k}=\mathrm e^{n\alpha+o(n)}$ for some $\alpha$ and that $\prod\limits_{k=1}^nb_{n-k}=\mathrm e^{n\beta+o(n)}$ for some $\beta$ and one considers $s_n=\sum\limits_{k=0}^{n-1}a_{n-1}\cdots a_{k+1}c_kb_{k-1}\cdots b_{0}$.
Let $\lambda\gt\max\{\alpha,\beta\}$, then $a_{n-1}\cdots a_{k+1}c_kb_{k-1}\cdots b_{0}\lt\mathrm e^{(n-k-1)\lambda+C}c_k\mathrm e^{k\lambda+C}=c_k\mathrm e^{n\lambda+C}$ where the value of $C$ may vary from one occurrence to the next. Since $\sum\limits_{k=0}^nc_k=O(n)$, this yields $s_n\leqslant\mathrm e^{n\lambda+C+O(\log n)}$, hence the Lyapunov exponent $\sigma$ of $(s_n)$ is such that $\sigma\leqslant\lambda$. This holds for every $\lambda\gt\max\{\alpha,\beta\}$ hence $\sigma\leqslant\max\{\alpha,\beta\}$.